The geometry of differential constraints for a class of evolution PDEs

TL;DR

This paper introduces a new constructive method for deriving differential constraints for evolution PDEs using generalized characteristics, with applications in stochastic filtering, soliton perturbations, and integrable systems.

Contribution

A novel method based on generalized characteristics for explicitly computing differential constraints in evolution PDEs is proposed.

Findings

Effective in deriving explicit differential constraints

Applicable to stochastic filtering and soliton equations

Validated through multiple detailed examples

Abstract

The problem of computing differential constraints for a family of evolution PDEs is discussed from a constructive point of view. A new method, based on the existence of generalized characteristics for evolution vector fields, is proposed in order to obtain explicit differential constraints for PDEs belonging to this family. Several examples, with applications in non-linear stochastic filtering theory, stochastic perturbation of soliton equations and non-isospectral integrable systems, are discussed in detail to verify the effectiveness of the method.